# There's a school of 1000 fish, the number P infected with a disease at time t years is given by P = #1000/(1+ce^-(1000t)# where c is a constant. How do you show that eventually all the fish will become infested?

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To show that eventually all the fish will become infested, we need to analyze the behavior of the function ( P = \frac{1000}{1+ce^{-1000t}} ) as ( t ) approaches infinity.

As ( t ) approaches infinity, the term ( e^{-1000t} ) will approach zero, since the exponential function ( e^{-1000t} ) decreases rapidly as ( t ) increases. Thus, the denominator of the fraction ( 1+ce^{-1000t} ) will approach 1.

As a result, the entire expression ( \frac{1000}{1+ce^{-1000t}} ) will approach ( 1000 ) as ( t ) approaches infinity.

Therefore, as time goes on (approaches infinity), the number of fish infected with the disease, represented by ( P ), will approach the total number of fish in the school, which is ( 1000 ). This indicates that eventually, all the fish in the school will become infested with the disease.

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